Localization of an Unstable Solution of a System of Three Nonlinear Ordinary Differential Equations with a Small Parameter

• Published in: Journal of applied and industrial mathematics 2022-11, Vol.16 (4), p.606-620
• Main Authors: Chumakov, G. A., Chumakova, N. A.
• Format: Article
• Language: English
• Subjects: Dynamical systems; Hopf bifurcation; Mathematical analysis; Mathematics; Mathematics and Statistics; Nonlinear differential equations; Nonlinear systems; Ordinary differential equations; Parameterization; Parameters; Subsystems
• ISSN: 1990-4789; 1990-4797
• DOI: 10.1134/S1990478922040032

In the present paper, we study some nonlinear autonomous systems of three nonlinear ordinary differential equations (ODE) with small parameter such that two variables are fast and the remaining variable is slow. In the limit as , from this “complete dynamical system” we obtain the “degenerate system,” which is included in a one-parameter family of two-dimensional subsystems of fast motions with parameter in some interval. It is assumed that there exists a monotone function that, in the three-dimensional phase space of a complete dynamical system, defines a parametrization of some arc of a slow curve consisting of the family of fixed points of the degenerate subsystems. Let have two points of the Andronov–Hopf bifurcation in which some stable limit cycles arise and disappear in the two-dimensional subsystems. These bifurcation points divide into the three arcs; two arcs are stable, and the third arc between them is unstable. For the complete dynamical system, we prove the existence of a trajectory that is located as close as possible to both the stable and unstable branches of the slow curve as tends to zero for values of within a given interval.